<PRE><B>Question 1181018</B>

Convert the General Equation 

{{{36y^2-64x^2-128x-144y-2512=0}}} to Standard form.

{{{(36y^2-144y) -(64x^2+128x)=2512}}}

{{{36(y^2-4y) -64(x^2+2x)=2512}}}

{{{36(y^2-4y+b^2) -36b^2-64(x^2+2x+b^2)-(-64)b^2=2512}}}

{{{36(y^2-4y+2^2) -36*2^2-64(x^2+2x+1^2)+64*1^2=2512}}}

{{{36(y-2)^2 -144-64(x+1)^2+64=2512}}}

{{{36(y-2)^2 -64(x+1)^2-80=2512}}}

{{{36(y-2)^2 -64(x+1)^2=2512+80}}}

{{{36(y-2)^2 -64(x+1)^2=2592}}}....both sides divide by {{{2592}}}

{{{36(y-2)^2/2592 -64(x+1)^2/2592=2592/2592}}}...simplify

{{{(y-2)^2/72 -2(x+1)^2/81=1}}}

{{{(y-2)^2/72 -(x+1)^2/(81/2)=1}}}

Up-down hyperbola with:

=&gt;{{{h=-1}}}, {{{k=2}}}, 
{{{a=sqrt(72)=6sqrt(2)}}}, {{{b=sqrt(81/2)=9/sqrt(2)}}}

Sketch and determine the parts of the Hyperbola.

Parts of Hyperbola:

1. Center:=&gt;{{{h=-1}}}, {{{k=2}}}=&gt; center ({{{-1}}}, {{{2}}})

2. Foci: {{{h, k+c)}}} or {{{h,k -c}}}

{{{c=sqrt(72+81/2)}}}-&gt;{{{c=15/sqrt(2)}}}

so, foci are at
({{{-1}}}, {{{2+15/sqrt(2))}}}) or ({{{-1}}}, {{{2-15/sqrt(2)}}})
({{{-1}}}, {{{12.6}}}) or ({{{-1}}}, {{{-8.6}}})

3. Vertices:({{{h}}}, {{{k+a}}}), and the other is at ({{{h}}}, {{{k-a}}})
({{{-1}}}, {{{2+6sqrt(2)}}}), ({{{-1}}}, {{{2-6sqrt(2)}}})
({{{-1}}}, {{{10.5}}}), ({{{-1}}}, {{{-6.5}}})

Co-vertices :
({{{-1+9sqrt(2)/2}}}, {{{2}}}) =&gt; ({{{5.4}}}, {{{2}}}) 
({{{-1-9sqrt(2)/2}}}, {{{2}}}) =&gt;({{{-7.4}}}, {{{2}}})

4. End Points of Conjugate Axis B1, B2
The co-vertices of a hyperbola are the endpoints of the conjugate axis
B1=({{{5.4}}}, {{{2}}})
B2=({{{-7.4}}}, {{{2}}})

distance between: {{{12.8}}}

5. End Points of Latus Rectum 
take y coordinate of the focus, as a line

foci is at ({{{-1}}}, {{{12.6}}}) or ({{{-1}}}, {{{-8.6}}})

take {{{x=-1}}} and find intersection with {{{(y-2)^2/72 -(x+1)^2/(81/2)=1}}}
{{{(12.6-2)^2/72 -(x+1)^2/(81/2)=1}}}

{{{x = -1 - 27/(4 sqrt(2))}}}≈ {{{-5.8}}}
{{{x = 27/(4 sqrt(2)) - 1}}}≈ {{{3.8}}}

E1= ( {{{-5.8}}},{{{12.6}}})
E2=({{{3.8}}},{{{12.6}}})
E3=({{{-5.8}}},{{{-8.6}}})
E4=({{{3.8}}},{{{-8.6}}})

6. Asymptotes: {{{y = 2/3 - (4x)/3}}},{{{ y = (4x)/3 + 10/3}}}
7. Eccentricity:{{{e = c/a=(15/sqrt(2))/(6sqrt(2))=5/4}}}
8. Length of LR:{{{2b^2/a=(2(9/sqrt(2))^2)/(6sqrt(2))=27/(2 sqrt(2)) ≈ 9.5
9. Length of Conjugate Axis:{{{9sqrt(2)}}}≈{{{12.7}}}
10. Length of Traverse Axis:12sqrt(2) ≈16.97






{{{drawing ( 600, 600, -15, 15, -15, 15,
circle(-1,2,.13),locate(-1.4,2,C),
circle(-1, 2+15/sqrt(2),.13),locate(-1.4, 2+15/sqrt(2),F),
circle(-1, 2-15/sqrt(2),.13),locate(-1.4, 2-15/sqrt(2),F),
circle(-1,2+10.5,.13),locate(-1,10.5,v),
circle(-1,-6.5,.13),locate(-1,-6.5,v),
circle(5.4,2,.13),locate(5.4,2,B1),
circle(-7.4,2,.13),locate(-7.4,2,B2),
circle(-5.8,12.6,.13),locate(-5.8,12.6,E1),
circle(3.8,12.6,.13),locate(3.8,12.6,E2),
circle(-5.8,-8.6,.13),locate(-5.8,-8.6,E3),
circle(3.8,-8.6,.13),locate(3.8,-8.6,E4),
blue(line(-5.8,12.6,3.8,12.6)), blue(line(-5.8,-8.6,3.8,-8.6)),
graph( 600, 600, -15, 15, -15, 15,- sqrt(72(1+(x+1)^2/(81/2)))+2,sqrt(72(1+(x+1)^2/(81/2)))+2,2/3-(4x)/3, (4x)/3 + 10/3)) }}} 


</PRE>